Exact Solutions and Stability Thresholds for the Fractional Gardner Equation with High-Order Dispersion

This study investigates the fractional Gardner equation with high-order dispersion, a fundamental model for nonlinear wave propagation in plasmas, optical fibers, and fluid systems. Using the modified extended direct algebraic (mEDA) method, we derive exact analytical solutions including bright/dark solitons, singular waves, and periodic patterns. All solutions have beenverified numerically using Mathematica to ensure their validity, as process innovation. The analysis reveals that the fractional order β significantly influences wave decay rates and memory effects, while specific parameter constraints govern solution existence and stability. A comprehensive linear stability analysis examines the modulation…